Exam 13: A: Modeling Computation

Suppose V={S, A, a, b}, T={a, b} , and S is the start symbol. Find a set of productions that includes $S \rightarrow A a$ and $A \rightarrow a$ and generates the language {a, b, b a, b a a} .

(Short Answer)

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Question 21

Let G be the phrase-structure grammar with vocabulary V={A, B, a, b, S} , terminal element set T={a, b} , start symbol S , and production set P= $\{ S \rightarrow A B a , S \rightarrow B a , A \rightarrow a B , A B \rightarrow b , B \rightarrow a b \}$ - Which of these are derivable from ABa? (1) ba , (2) abb , (3) aba , (4) b , (5) aababa

(Short Answer)

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Question 22

Let G be the phrase-structure grammar with vocabulary V={A, B, a, b, S} , terminal element set T={a, b} , start symbol S , and production set P= $\{ S \rightarrow A B a , S \rightarrow B a , A \rightarrow a B , A B \rightarrow b , B \rightarrow a b \}$ are derivable from A ? (1) babaa, (2) aab , (3) bba

(Short Answer)

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Question 23

Determine if 1101 belongs to the regular set 101.

(Short Answer)

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Question 24

Consider the Turing machine T : $\left( s _ { 0 } , 0 , s _ { 1 } , 1 , R \right) , \left( s _ { 0 } , 1 , s _ { 1 } , 1 , R \right) , \left( s _ { 1 } , 0 , s _ { 0 } , 1 , L \right) , \left( s _ { 1 } , 1 , s _ { 0 } , 0 , R \right) , \left( s _ { 0 } , B , s _ { 1 } , 1 , R \right)$ For the following tape, determine the ﬁnal tape when T halts, assuming that T begins in state s0 at the leftmost nonblank symbol. \@cdots 1 0 1 \@cdots

(Short Answer)

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Question 25

Find a deterministic ﬁnite-state automaton equivalent to the following nondeterministic ﬁnite-state machine.

(Essay)

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Question 26

Let A={0,11} . Find A² .

(Short Answer)

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Question 27

determine the output for each input string, using -000

(Short Answer)

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Question 28

Let G be the phrase-structure grammar with vocabulary V={A, B, 0,1, S} , terminal elements T={0,1} , start symbol S , productions P= $\{ S \rightarrow A B 0 , A B \rightarrow 1 , A \rightarrow 0 , B \rightarrow A B \}$ from S ? (1) 000, (2) 11, (3) 010, (4) } 0000, (5) 0001, (6) 110, (7) 0010

(Short Answer)

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Question 29

Suppose a phrase-structure grammar has productions S → 1S0, S → 0A, A → 0. Find a derivation of

(Short Answer)

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Question 30

Find a set of two productions that produces $\left\{ 1 ^ { 2 n } \mid n > 0 \right\}$

(Short Answer)

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Question 31

Find all strings recognized by this deterministic ﬁnite-state automaton.

(Short Answer)

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Question 32

Find the Kleene closure of A={1} .

(Short Answer)

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Question 33

Suppose a phrase-structure grammar has productions S → S0, S → A1, A → 0. Find a derivation of 01.

(Short Answer)

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Question 34

determine the output for each input string, using -10111

(Short Answer)

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Question 35

Determine if 1101 belongs to the regular set (0∪1)*1.

(Short Answer)

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Question 36

Suppose that A = {1, 11, 01} and B = {0, 10}. Find AB.

(Short Answer)

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Question 37

Construct a ﬁnite-state automaton that recognizes the set represented by the regular expression 10∗.

(Essay)

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Question 38

Which strings belong to the set represented by the regular expression 0∗ ∪ 11?

(Essay)

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Question 39

Find the language recognized by this nondeterministic ﬁnite-state automaton.

(Short Answer)

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Question 40

Exam 13: A: Modeling Computation

Suppose V={S, A, a, b}, T={a, b} , and S is the start symbol. Find a set of productions that includes S→AaS \rightarrow A aS→Aa and A→aA \rightarrow aA→a and generates the language {a, b, b a, b a a} .

Determine if 1101 belongs to the regular set 1*0*1.

Find a deterministic ﬁnite-state automaton equivalent to the following nondeterministic ﬁnite-state machine.

Let A={0,11} . Find A2 .

determine the output for each input string, using -000

Suppose a phrase-structure grammar has productions S → 1S0, S → 0A, A → 0. Find a derivation of

Find a set of two productions that produces {12n∣n>0}\left\{ 1 ^ { 2 n } \mid n > 0 \right\}{12n∣n>0}

Find all strings recognized by this deterministic ﬁnite-state automaton.

Find the Kleene closure of A={1} .

Suppose a phrase-structure grammar has productions S → S0, S → A1, A → 0. Find a derivation of 01.

determine the output for each input string, using -10111

Determine if 1101 belongs to the regular set (0∪1)*1.

Suppose that A = {1, 11, 01} and B = {0, 10}. Find AB.

Construct a ﬁnite-state automaton that recognizes the set represented by the regular expression 10∗.

Which strings belong to the set represented by the regular expression 0∗ ∪ 11?

Find the language recognized by this nondeterministic ﬁnite-state automaton.

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

Suppose V={S, A, a, b}, T={a, b} , and S is the start symbol. Find a set of productions that includes $S \rightarrow A a$ and $A \rightarrow a$ and generates the language {a, b, b a, b a a} .

Determine if 1101 belongs to the regular set 101.

Let A={0,11} . Find A² .

Find a set of two productions that produces $\left\{ 1 ^ { 2 n } \mid n > 0 \right\}$